You have not shown a series, but a sequence of partial sums. These are two quite-different things. The values in the sequence of partial sums do "approach" the value "1". And this sequence, along with it's limit value, is used to prove, according to the definition, that the associated (but different) series has a value of 1 -- not that the series "approaches" 1, because series do not "approach" anything, but that the value of the infinite sum is "1".

Please take some time to study the differences between sequences and series, to learn the definition of a limit, and to learn what a limiting procedure is. Once you understand the terms and processes that are under discussion, you will likely have a better chance at understanding the answers that have been given.

Thank you.